# law of one price, understanding

I am reading about mathematical finance, and I was tipsed to ask the quesiton on this site. It is about the "law of one price".

Just first I'll make precise the model my book uses:

I have a single period, so I only have time t=0 and t=1.

$B_t$ is the bank account process, where $B_0=1$, and $B_1 \geq1$ is a stochastic variable.

The price process is $S=\{S_t: t=0,1\}$, where $S_0=(S_1(0),,.S_N(0))$ is the starting price for each security, and $S_1=(S_1(1),,...,S_n(1))$ are stochastic variables giving the end price.

$H=(H_0,..H_n)$ gives the trading strategy($H_0$ is just the starting money in the bank, and each $H_i$ is just the number of shares). So the value(it is a stochastic variable when t=1) is:

$V_t=H_0*B_t+\Sigma H_i*S_n(t)$.

This is the model the book uses.

Now comes the definition of the "law of one price": However I struggle to see why this is intuitive. I know that if this law don't hold we have arbitrage or a dominant strategy, so I've seen explanations that says if the law of one price doesn't hold, then we have arbitrage, and hends it is an illogical market.

However, I am wondering if the point of the "law of one price" can be explained without using "arbitrage" or "dominant trading strategies".

For instance like this figure shows, we may have two markets where we have arbitrage or dominant trading strategy, but where in one we have the law of one price, and in the other we don't have the law. How would you in this case explain that the law of one price gives a more realistic market?(you can not explain it with arbitrage or dominant trading strategy in this case). PS: I know very little about finance terms, so i would really appreciate it if you explained in terms of the model I wrote in the start.

From my knowledge, Law of One Price is defined as:

If two assets provide the same cashflows, they must have the same price.

This is the justification to price options by a replicating portfolio.

The model here seems to assume some European Claims 1-period model, which means $V_1$ represents a final payoff. At the (only) prior time $t=0$, the values of two claims with the same payoffs must by LOP have the same price, which means no $V_0>V_0'$.

So for option pricing, one can create a portfolio of bond and stock which end with the same value as the payoff of a European option. Hence by LOP, the option must be worth same as the replicating portfolio.

This is not exactly the same as saying there was no arbitrage, because it might be the case that a replicating portfolio does not exist (incomplete market), while the market itself is still arbitrage-free (when a riskneutral measure Q exsists).

• Thank yoiu very much. But why is it that in the two markets market 1 and market 2, market 1 is considered more realistic? Why is it unrealistic for two persons to start of with a different amount of money but certainly end up with the same? – user119615 Jul 31 '14 at 16:33
• @user119615 If you think of it as traded assets, if one asset would provide same cashflows for a lower price, the market would buy this asset until its price reaches LOP. Certainly, if one person holds an asset which it can sell for more, he will. – emcor Jul 31 '14 at 16:37
• Thanks, I think i almost understand it now, but I am not quite there, sorry if these are stupid questions. When you say asset you mean the enitre collection ($H_0$,$H_1$,..$H_N$)?, together they are an asset? Also why would the market want to buy them? I mean the definition only says that one trading strategy costing more than another trading-strategy would will give the same value at the end. So if there was only these two possibilities I understand that everybody would buy the cheapest one. But there are endless possibilities for the market?, and there might be other things the market finds – user119615 Jul 31 '14 at 16:59
• more favourable? The thing is, I understand why in the situation of arbitrage or dominant trading strategy, the market would buy up all of those things, but in the case of not having the law of one price, I don't see why people automatically would rush to buy the cheaper of the two, just because in the end they are worth the same. I mean, just because one deal is better than another deal, it doesn't mean it is a good deal? – user119615 Jul 31 '14 at 17:00
• Wow, that's a lot of comments. @emcor and user119615 if you could put some of this in your answer and/or the question, that would be great. It's a bit hard to follow now. – Bob Jansen Jul 31 '14 at 19:12

Qian's (2011) book says:

The law of one price (LOP) states that portfolios with the same payoff must have the same price:

$X' h = X' h \Rightarrow p' h = p' \tilde h$

where $p \in \mathbb{R}^J$ is the price vector.